Question

show that any positive even integer is of form 8m or 8m+2 or 8m+4 or 8m+6 (use Euclid's division lemma)

Answer 1

Let b=8,this implies r=0,1,2,3,4,5,6,7
If r=0,
a=bq+r
a=8q+0
By squaring on both sides, we get,
a^2 =(8q+0)^2
a2=64q2
a2=8(8q2)
a2=8m (where m=8q2)



Use same method for other proofs....


Hope it helps you.....

Answer 2

Answer:

Step-by-step explanation:

let a and b be any positive interger such that a>b

then by  Euclid division lemma

a=bq + r

where b=4 and 0≤r<b

therefore

a=4q+0  ,   a=4q+1   ,   a=4q+2  ,  a=4q+3  

consider a=4q+1

              a²=( 16q² +8q +1)

              a²=8(2q² +q) +1

let m be = 2q² +q  

 therefore, a²=8m+1

 the square of any positive inter is of form 8m+1 for some integer m . ​